Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

We derive the exact asymptotics of $P\left( \sup_{t\ge 0} \Bigl( X_1(t) - μ_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - μ_2 s\Bigr)> u \right), \ \ u\to\infty,$ where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional Brownian motion with correlation $ρ\in[-1,1]$ and $μ_1,μ_2>0$. It appears that the play between $ρ$ and $μ_1,μ_2$ leads to several types of asymptotics. Although the exponent in the asymptotics as a function of $ρ$ is continuous, one can observe different types of prefactor functions depending on the range of $ρ$, which constitute a phase-type transition phenomena.

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